It is important in the proof of Theorem REF  that the dependence of the estimate in Theorem REF  on the coefficients \(a_1,\ldots , a_4\)  is made completely explicit.
Several authors have obtained estimates of this type for quadratic forms. In [1]}, Browning applies the machinery from [2]} to find such an estimate for the counting problem corresponding to \(N_{\mathbf {a}}(B)\) , but in \(n\geqslant 5\)  variables. Subsequently in [3]}, Browning and Heath-Brown carried this out in four variables. This latter result is nearly sufficient for our purposes, but Theorem REF  represents a refinement in which we drop the assumptions made in [3]} that \(A\)  is nearly square-free and all the coefficients \(a_1,\ldots , a_4\)  are roughly the same size. Finally, we mention that using other techniques from the geometry of numbers, Comtat provides in [5]} a completely uniform estimate for the number of zeros \((x_1,\ldots ,x_n)\in \mathbb {Z}^n_{\textrm {prim}}\)  of a non-singular quadratic form \(Q\)  in \(n\geqslant 3\)  variables which lie in an arbitrary box \(|x_i|\leqslant B_i\)  for \(i\in \lbrace 1,\ldots , n\rbrace \) . However, the resulting bound \(N_{\mathbf {a}}(B) \leqslant B/|A|^{1/4}\)  does not have a good enough dependence on \(A\)  to be useful for our purposes.
